Hedging by Sequential Regression an Introduction to the Mathematics of Option Trading

8/3/2019 Hedging by Sequential Regression an Introduction to the Mathematics of Option Trading

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H E D G I N G B Y S E Q U E N T I A L R E G R E S S IO N : A N I N T R O D U C T I O N T OT H E M A T H E M A T I C S O F O P T I O N T R A D I N G

BY H . FOLLMER a n d M . SCHW EIZERETH Ziirich

1. INTRODUCTIONI t i s w i d e l y a c k n o w l e d g e t h a t t h e r e h a s b e e n a m a j o r b r e a k t h r o u g h i n t h em a t h e m a t i c a l t h e o r y o f o p t i o n t r a d i n g . T h i s b r e a k t h r o u g h , w h i c h is u s u a l ly s u m -m a r i z e d b y t he B l a c k - S c h o l e s f o r m u l a , h a s g e n e r a t e d a l ot o f e x c i te m e n t a n d ac e r t a i n m y s t i q u e . O n t h e m a t h e m a t i c a l s i d e , i t i n v o l v e s a d v a n c e d p r o b a b i l i s t i ct e c h n i q u e s f r o m m a r t i n g a l e t h e o r y a n d s t o c h a s t i c c a l c u l u s w h i c h a r e a c c e s s i b l eo n l y t o a s m a l l g r o u p o f e x p e r t s w i t h a h i g h d e g r e e o f m a t h e m a t i c a l s o p h i s t i -c a t io n ; h e n c e th e m y s t iq u e . I n i ts p r a c t i c a l imp l ic a t io n s i t o f f e r s e x c i tin g p r o s p e c t s .I t s p r o m i s e i s t h a t , b y a s u i t a b l e c h o i c e o f a t r a d i n g s t r a t e g y , t h e r i s k i n v o l v e di n h a n d l i n g a n o p t i o n c a n b e e l i m i n a t e d c o m p l e t e l y .

S i n c e O c t o b e r 1 9 8 7 , t h e m o o d h a s b e c o m e m o r e s o b e r . B u t t h e r e a r e a l s om a t h e m a t i c a l r e a s o n s w h i c h s u g g e st t h a t e x p e c t a t i o n s s h o u l d b e l o w e r e d . T h i sw i ll b e t h e m a i n p o i n t o f t h e p r e se n t e x p o s i t o r y a c c o u n t . W e a r g u e t h a t , t y p i c a l l y ,th e r i sk in v o lv e d in h a n d l in g a n o p t io n h a s a n i r r e d u c ib le in t r in sic p a r t . Th i s in t r in -s ic ri s k m a y b e m u c h s m a l l e r t h a n t h e a priori r i s k , b u t i n g e n e r a l o n e s h o u l d n o te x p e c t it t o v a n i s h c o m p l e t e l y . I n t h i s m o r e s o b e r p e r s p e c t i v e , t h e m a t h e m a t i c a lt e c h n iq u e b e h i n d t h e B l a c k - S c h o l e s f o r m u l a d o e s n o t l o se a n y o f it s i m p o r t a n c e .I n f a c t , i t s h o u ld b e s e e n a s a s e q u e n t ia l r e g r e s s io n s c h e me wh o s e p u r p o s e i s tor e d u c e t h e a priori r i s k to i t s in t r in s ic c o r e .

W e b e gi n w i th a s h o r t i n t ro d u c t i o n t o t h e B l a c k - S c h o l e s f o r m u l a i n t e r m s o fc u r r e n c y o p t i o n s . T h e n w e ci ev e lo p a g e n e r a l r e g r e s s io n s c h e m e i n d i sc r e te t i m e ,f ir st i n a n e l e m e n t a r y t w o - p e r i o d m o d e l , a n d t h e n i n a m u l t i p e r i o d m o d e l w h i c hi n v o lv e s m a r t i n g a l e c o n s i d e r a t i o n s a n d s e ts th e s t a g e f o r e x te n s i o n s t o c o n t i n u o u st im e . O u r m e t h o d is b a s e d o n t h e i n t e r p r e t a t io n a n d e x t e n s io n o f t h eB l a c k - S c h o l e s f o r m u l a i n t e r m s o f m a r t i n g a l e t h e o r y . T h i s w a s in i ti a te d b y K r e p sa n d H a r r i s o n ; s e e , e .g . th e e x c e ll e n t s u r v e y o f HARRISON a n d PLI SKA ( 1 9 81 , 1 9 8 3) .T h e i d e a o f e m b e d d i n g t h e B l a c k - S c h o l e s a p p r o a c h i n to a s e q u e n ti a l r e g re s s io ns c h em e g o e s b a c k t o j o in t w o r k o f t h e f ir st a u t h o r w i t h D . S o n d e r m a n n . I n co n -t i n u o u s t i m e a n d u n d e r m a r t i n g a l e a s s u m p t i o n s , t h is w a s w o r k e d o u t i nSCHWEIZER (1984) a nd FOLLMER an d SONDERMA NN (1986) . SCHWEIZER (1988)d e a l s w i t h t h e s e p r o b l e m s i n a g e n e r a l s e m i m a r t i n g a l e m o d e l .

T h e p r e s e n t p a p e r i s a w r i t t e n v e r s i o n , w i t h s o m e e x t e n s i o n s , o f a n e x p o s i t o r yt a l k g i v e n a t t h e a n n u a l m e e t i n g o f t h e V e r e i n i g u n g S c h w e i z e r i s c h e r V e r -s i c h e r u n s m a t h e m a t i k e r i n S e p t e m b e r 1 98 7. A s in t h e t a l k , o u r p u r p o s e is t o p r o -v i de a n e le m e n t a r y i n t r o d u c t i o n t o s o m e k e y f e a t u re s o f t h e m a t h e m a t i c a l t h e o r yo f o p t i o n p r i c i n g , w i t h s p e c ia l e m p h a s i s o n t h e u s e o f l in e a r r e g r e s s io n .A ST I N B ULLETIN, VoI . 19 , S


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30 FOLLMER AND SCHWEIZER2. FROM HUYGENS AND BERNOULLI TO THE FORMULA OF BLACK AND SCHOLES

C o n s i d e r a c a l l o p t i o n o n U S d o l l a r s a g a i n s t S w i s s f r a n c s . T h i s o p t i o n g i v e s t h er i gh t t o buy a spec i f i ed amoun t o f do l l a r s , s ay $100 , a t a spec i f i ed t ime T a t ap r e d e t e r m i n e d e x c h a n g e r a t e K . I f Xt deno t e s t he exchange r a t e , i . e . t he va luei n S F R o f $ 1 0 0 a t ti m e t E [ 0 , T ] , t h e n t h e v a l u e Vr o f t h e o p t i o n a t t h e t e r m i n a l

" t i rne T wi l l beH = ( X r _ K ) + . [ 0 X r - K i f X r ~ > Ko t h e r w i s e .

I t is n o w n a t u r a l t o a s k " W h a t i s a f a i r p ri c e f o r t h is o p t i o n ? I n o t h e r w o r d s : W h a tis the va lue Vo o f t he op t i on a t t he i n i t i al t im e 0 when t he f i na l ou t co m e H i s s ti llu n c e r t a i n ?U n t i l 1 9 73 , w h e n t h e f u n d a m e n t a l p a p e r s o f B L A C K a n d S C HO L ES ( 1 97 3 ) a n dM E R TO N ( 1 97 3 ) a p p e a r e d , t h e r e s e e m e d t o b e a n o b v i o u s a n s w e r . T h i s a n s w e rcou ld have been g iven a l r ea dy by CH R. HUYGENS ( 1657 ) and BERNOULLI ( 1713 ).T o b e g i n w i th , t h e e x c h a n g e r a t e X r a n d , c o n s e q u e n t l y , t h e r e t u r n H o f th eo p t i o n s h o u l d b e v i e w e d a s r a n d o m v a r i a b l e s o n s o m e p r o b a b i l i t y s p a c e ( f] , 0~-, P )w h i c h d e s c r i b e s t h e p o s s i b l e t i m e e v o l u t i o n s o f t h e e x c h a n g e r a t e a n d t h e i rr e spec t ive p r obab i l i t i e s . G iven such a s t ochas t i c m ode l , t h e f a i r p r i c e o f t heo p t i o n s h o u l d b e e q u a l t o t h e e x p e c t e d v a l u e E [ H ] o f t h e r a n d o m v a r i a b l e H :(2 .1) Vo = E[ H] .O n e c o u l d t h i n k o f se v e r a l m o d i f i c a ti o n s . F o r e x a m p l e , o n e c o u l d t a k e i n t oaccoun t an i n t e r e s t r a t e r and r ep l ace ( 2 .1 ) by

1(2.2) V0 = ~ " E [ H ] .l + rA l s o , t h e p r i c e c o u l d i n c l u d e a r i s k p r e m i u m , a n d i t w o u l d s e e m r e a s o n a b l e t oc o m p u t e s u c h a p r e m i u m in te r m s o f t h e v a ri a n c e V a r [ H ] o f t h e r a n d o m v a r ia b l eH , s i n c e t h i s v a r i a n c e w o u l d a p p e a r t o b e a n a t u r a l m e a s u r e o f t h e r i s k i n v o l v e di n h a n d l i n g t h e o p t i o n . I n e s s e n c e , h o w e v e r , t h e p r o b l e m w o u l d h a v e s e e m e d t ob e r e d u c e d t o t h e c h o i ce o f a s u it a b l e p r o b a b i l i t y m e a s u r e P .S u c h a p r o b a b i l i t y m e a s u r e P o n t h e s p a c e f~ o f a l l c o n t i n u o u s t i m e e v o l u t i o n so~: [ 0 ,1 ] ~ R w as p r o po sed i n 1900 by Bache l i e r i n h is t he si s ' T h6o r i e de l aS p 6 c u l a t i o n ' ; a r i g o r o u s c o n s t r u c t i o n w a s g iv e b y W i e n e r i n 1 9 23 . U n d e r t h isW iene r m easu r e P , t he s t ochas t i c p r oces s X / ( ~ , ) := o~ ( t) ( 0 ~< t ~ T ) b ehav es l i kea B r o w n i a n m o t i o n . T h i s s t o c h a s t i c m o d e l h a s t u r n e d o u t t o b e o f b a s i c i m p o r -t a n c e , n o t o n l y f o r i t s f u n d a m e n t a l c o n n e c t i o n t o p h y s i c s , b u t a l s o o n p u r e l ym a t h e m a t i c a l g r o u n d s . I t i s , f o r e x a m p l e , t h e n a t u r a l r e f e r e n c e m o d e l f o r f u n c -t i ona l v e r s i o n s o f t h e c en t r a l li m i t t h e o r e m .

I n v i e w o f a p p l i c a t i o n s t o t h e s t o c k m a r k e t o n e m i g h t w a n t t o u s e a m o d i f i e dv e r s io n . F o r e x a m p l e , S A M UE LS O N ( 19 6 4 ) h a s p r o p o s e d t o m o d e l t h e s t o c h a s t i cp r oces s X t ( 0 ~< t ~< T ) a s t he so lu t i on o f a s t och as t i c d i f f e r en t i a l e qu a t i on(2 .3) d Xt = o . Xt dBt + # " Xt dt


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HEDGING BY SEQUENTIAL REGRESSION 31w h e r e ( B t ) is a B r o w n i a n m o t i o n . H e r e t~ " X t d t i s a t r e n d f o r e c a s t w i t h d r i f tp a r a m e t e r t ~ , a n d o . X t d Bt is a r a n d o m f l u c t u a t i o n w i t h v o l a t il i t y p a r a m e t e r o .S u p p o s e n o w t h a t w e a cc e p t ( 2 .3 ) a s a m a t h e m a t i c a l m o d e l f o r t h e s t o c h a s t i c ti m ee v o l u t i o n o f o u r e x c h a n g e r a t e X t ( 0 ~< t ~< T ) . I n t h i s m o d e l , t h e r a n d o mv a r i a b l e X r h a s a lo g - n o r m a l d i s t r ib u t i o n , a n d t h e H u y g e n s - B e r n o u l l i p r e s c r ip -t i o n ( 2 . 1 ) w o u l d l e a d t o f o r m u l a( 2 . 4 )

V o = E [ H ]= I2 ~ ( o . , f T . + ( t ~ - ~ tr 2 ) T ) - K ) e x p ( - - ~ ) d u .2 ( o e x , u ' +

B u t i n t h e s a m e m o d e l ( 2 . 3 ) , t h e B l a c k - S c h o l e s f o r m u l a g i v e s a q u i t e d i f f e r e n ta n s w e r . I t te ll s y o u t h a t t h e d r i f t p a r a m e t e r tz is c o m p l e t e l y i r r e le v a n t , t h a t y o um i g h t a s w e l l r e p la c e P b y t h e m e a s u r e P * c o r r e s p o n d i n g t o tz * = 0 w h i c h m a k e st h e e x c h a n g e r a t e b e h a v e l i k e a f a i r g a m e , a n d t h a t t h e f a i r p r i c e o f t h e o p t i o ns h o u l d b e c o m p u t e d a s t h e e x p e c t e d v a l u e o f H i n t h i s n e w m o d e l , e v e n t h o u g ht h e e x p e r t s ' f o r e c a s t is g i v e n b y P a n d n o t b y P * :( 2 . 5 )

V o = E * [ H ]1 " I ~ ( ( " ~ )= 2 ~ X o " e x p (t r. 4 ~ . u - l " tr2 " T ) - K ) + e xp - d u

= X o o ' , , ~ " l o g - ~ - - K ' , I , o -~ fT " lo g K - 2 'w h e r e ~ d e n o t e s t h e c u m u l a t i v e d i s tr i b u t io n f u n c t i o n o f a s ta n d a r d n o r m a ld i s t r i b u t i o n N ( 0 , 1 ). A l s o , i t i s c l a i m e d t h a t t h e r e i s a t r a d i n g s t r a t e g y w h i c h r e -q u i r e s t h e i n i ti a l i n v e s t m e n t V 0 = E * [ H ] a n d t h e n d u p l i c a t e s t h e c o n t i n g e n t c l a i mH ( ~ ) w i t h o u t a n y a d d i t i o n a l c o s t , n o m a t t e r w h i c h t i m e e v o l u t i o n 0 E f l i s r e a l-i ze d b y t h e r a n d o m m e c h a n i s m d e s c r i b e d b y P . I n p a r t ic u l a r , t h e r e is n o r e a s o nt o m o d i f y t h e f a ir p ri c e V o = E * [ H ] b y a r is k p r e m i u m .

I f w e i n c l u d e i n t e r e s t r a t e s i n t h e m o d e l , t h e n ( 2 . 5 ) i s r e p l a c e d b y( 2 . 6 ) Vo = E *[ ( X r - K . e - " 7")+ ]

= X o ~ ( o 1 " ( l o g X o 1

- - K - e - # ' r - o ( - o 1 - ( l o g X o 1

H e r e 0 d e n o t e s t h e c o n t i n u o u s l y c o m p o u n d e d r a t e o f r e t u r n o n a S w i s s f r a n ca c c o u n t .


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32 FOLLMER AND SCHWEIZER

A r igorous mathemati cal justification of this counter-intuitive prescription in-volves rather advanced tools, e.g. the Cameron-Martin-Maruyama-Girsanovformula an d a deep representa tion theorem of K. ITt3 (1951). This explains someof the mystique around the Black-Scholes formula. But from an economic pointof view, its crucial feature can already be explained in a very elementary settingwhich does not require any mathe matical sophistication. To this end, we employthe well-known didactic device of using an example with binary structure.Suppose that the current exchange rate is given by X0 = 135. Consider a callopt ion with a strike of K = 145 at time T. We assume the following binaryscenario: the exchange rate at time T will be 175 with prob abil ity p or 80 withprobabilit y 1 - p. Correspon dingly, the return H of the opt ion will be 30 withprobab ility p and 0 with proba bility 1 - p. Taking into account an interest rater, the Huygens-Bernoulli prescription (2.2) would compute the fair price of theoption as(2.7) Vo = - 1'---~" p" 30;for p = 0.5 and r = 0.05 we would get V0 = 14.3. The Black-Scho les prescr iption,however, wou ld be the following. First replace p by p* so that the exchange rate,properly discounted, behaves like a fair game:

or, more explicitly,, r x r lXo-- Ll+rJ

1135 = l + r ( p * . 1 7 5 + ( 1 - p * ) - 8 0 ) .Now comput e the fair price as the expected value of the return H, properly dis-counted, in this new model:

E*[ H ] 1 .p *. 30 ;Vo= Ll+r j= l+rfor r = 0.05 we would get p * = 0.65 and V0 = 18.6.

At first sight, this change of the model seems completely arbitrary, just as inthe more int ricat e model above. But in the present simple case we can give a directeconomic justification. Suppose that at time 0 you sell the opti on. Then you canprepare for the resulting contingent claim at time T by using the followingstrategy:

Sell the option at the Black-Scholes price 18.6Buy $31.6 at the present exchange rate of 1.35Take a loan of SFR 24.1 with interest ra te r = 0.05

+ 18.6- 42.7+24.1

0


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HEDGING BY SEQUENTIAL REGRESSION 33T h u s , t h e b a l a n c e a t t i m e 0 is 0 . A t t i m e T w e h a v e t o d i s t i n g u i s h t w o c a s e s :

( i ) T h e d o l l a r h a s r i s e n : O p t i o n is e x e r ci z ed - 3 0Se l l do l l a r s a t 1 . 75 + 55 . 3P a y b a c k l o a n w i t h i n te r e s t - 2 5 . 3

0( ii ) T h e d o l l a r h a s f a l l e n : O p t i o n e x p i r e s 0

S e l l d o l l a r s a t 0 . 8 0 + 2 5 . 3P a y b a c k l o a n w i t h i n te r e s t - 2 5 . 3

0T h i s d e m o n s t r a t e s t h a t t h e B l a c k - S c h o l e s p r ic e 1 8 .6 is j u s t t h e r i g h t a m o u n tw h i c h is n e e d e d i n o r d e r to h e d g e t h e o p t i o n w i t h o u t a n y r i sk . A n y o p t i o n p r i c ed i f fe r e n t f r o m t h e B l a c k - S c h o l e s p r ic e w o u l d e n a b l e e i th e r th e o p t i o n s e ll er o rt h e o p t i o n b u y e r to m a k e a s u r e p r o f it w i t h o u t a n y r i sk : T h e r e w o u l d b e a na r b i t r a g e o p p o r t u n i t y .

I n th e n e x t s e c ti o n , w e e x p l a in h o w t h e c o r r e ct h e d g i n g s t r a t e g y c a n b e f o u n di n a s y s t e m a t i c w a y . I n t h e p r e c e d i n g e x a m p l e i t i s c l e a r t h a t t h e m o d e l is t o os i m p l i s t ic : t h e r e i s n o r e a s o n t o r e s t r ic t o u r a t t e n t i o n t o a b i n a r y s c e n a r i o .T h e r e f o r e , w e a r e g o i n g t o a d m i t m o r e g e n e r a l m o d e l s . B u t t h is w i ll fo r c e u s tol o w e r o u r e x p e c t a t i o n s t o a m o r e r e a l is t i c l e v el . I n f a c t , t h e s i t u a t i o n b e c o m e s l e s sp l e a s a n t a s s o o n a s w e a d m i t a t h i r d p o s s i b i l i t y f o r t h e v a l u e o f X r : I t is n o l o n g e rp o s s i b l e t o r e d u c e t h e r i s k t o 0 .

F r o m t h e p o i n t o f v ie w o f th e c o n t i n u o u s - t i m e m o d e l ( 2 .3 ) , a b i n a r y s i tu a t i o na s a b o v e s h o u l d o n l y s er v e as a n i n f in i t e si m a l b u i ld i n g s t o n e a n d n o t b e t a k e ns e r i o u s ly in i ts e lf . B u t a ls o o n t h e m a t h e m a t i c a l l y m u c h m o r e a d v a n c e d l ev e l ( 2 .3 )t h e r e a re g o o d r e a s o n s t o b e li e v e t h a t t h e m o d e l , a n d i n p a r t ic u l a r i t s p r o m i s e o fr i s k - f re e o p t i o n t r a d i n g , i s t o o n i c e t o b e r e a l i s t ic .

3. HEDG ING IN A TWO-P ERIOD MODEL: AN EXERCISE IN LINEAR REGRESSIONL e t u s f ir s t c o n s i d e r a s i m p l e t w o - p e r i o d m o d e l w h e r e t h e e x c h a n g e r a t e s X k a tt h e i n i t ia l t i m e k = 0 a n d t h e t e r m i n a l t i m e k = 1 a r e r a n d o m v a r i a b l e s o n s o m ep r o b a b i l i t y s p a c e (f ~, ,~ , P ) . A t t i m e 0 t h e e x c h a n g e r a t e X o is k n o w n a n d c a n b et r e a t e d a s a c o n s t a n t , i . e. w e a s s u m e t h a t P [Xo = x 0 ] = 1 f o r s o m e x 0 > 0 . T h eo p t i o n , o r r a t h e r t h e r e s u l t in g c o n t i n g e n t c l a i m a t ti m e 1 , i s d e s c r i b e d b y a r a n -d o m v a r i a b l e H d e f in e d o n t h e s a m e p r o b a b i l i t y sp a c e . F o r a c a ll o p t i o n w i t h as t r ik e o f K , w e w o u l d h a v e H = (X ~ - K ) + . T o k e e p th e e x p o s i t i o n a s s i m p le a sp o s s i b l e , w e l e a v e i n t e r e s t r a t e s a s i d e f o r t h e m o m e n t .

L e t u s n o w a s s u m e t h a t w e h a v e s o ld t h e o p t i o n . A t t i m e 1 w e w i ll h a v e t o p a yt h e r a n d o m a m o u n t H ( o~ ). W e s h o u l d l ik e to i n s u r e o u r s e lv e s a g a i n s t t h is e v e nt ;w e w a n t t o hedge t h e o p t i o n . T o t h is e n d , w e b u y 1 0 0 . ~ d o l l a r s a n d p u t a s i d e


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3 4 FOLLMER AND SCHWE1ZERTo S w i s s f r a n c s . T h i s i n i t i a l p o r t f o l i o a t t i m e 0 h a s t h e v a l u e(3. I) Vo = 6" Xo + T/o.A t t h e t e r m i n a l t i m e 1 , w e w a n t a p o r t f o l i o w h o s e v a l u e i s e x a c t l y e q u a l t o H .T h e v a l u e o f t h e d o l l a r a c c o u n t w i ll b e 6 " X t , a n d i f w e t h e n a d j u s t t h e S w i ssf r a n c a c c o u n t f r o m ~/o t o 7/~ = H - 6 " X ~ , t h e v a l u e(3 .2) VI = ~- X l + v /lo f t h e r e s u l ti n g p o r t f o l i o a t t im e 1 w i ll sa t i s f y o u r c o n d i t i o n(3 .3) V1 = H .F o r a g i v e n H , s u c h a s t r a t e g y w il l b e d e t e r m i n e d b y o u r i n i t ia l c h o i c e o f t h e c o n -s t a n t s 6 a n d V o.

L e t u s e x a m i n e t h e c o s t s i n d u c e d b y s u c h a s t r a t e g y ( 6, V o ) . I f C k d e n o t e s t h ec u m u l a t i v e c o s t u p t o t i m e k , t h e n w e h a v e(3 .4) Co = Vo,a n d t h e a d d i t i o n a l c o s t d u e t o o u r a d j u s t m e n t o f t h e S w i s s f r a n c a c c o u n t a t t i m e1 i s g i v e n b y( 3 . 5 ) C i - Co = ~l - v/o

= - 6 " - ( V o - X o )= V ~ - V o - 6 . A X

w h e r e w e p u t A X . '= - X ~ - X o . L e t u s n o w c h o o s e o u r t r a d i n g s t r a t e g y ( 6, V o ) i ns u c h a w a y t h a t t h e r e m a i n i n g r i s k a t t im e 0 , m e a s u r e d b y t h e e x p ec t e d q u a d r a t i cCOSt( 3 . 6 ) R : = E [ (C ~ - C o ) 2 ]

= E [ ( H - Vo - 6 " A X ) 2 ] ,is m i n i m i z e d . T h i s is , o f c o u r s e, a w e l l - k n o w n p r o b l e m : w e a re s i m p l y l o o k i n gf o r t h e b es t l i n e a r e s t i m a t e o f H b a s e d o n A X . T h u s , t h e o p t i m a l c o n s t a n t s 6 a n dV o a r e g i v e n b y(3 .7) 6 =a n d( 3 . 8 )

C o y ( H , A X ) = C o y ( H , X t )V a r [ A X ] V a r [ X ~ ]

It"o = E [ H ] - 6 " E [ A X ] .T h i s o p t i m a l v a l u e V0 o f t h e i n i ti a l p o r t f o l i o m a y b e r e g a r d e d a s a f a i r p r i c e o ft h e o p t i o n . I n p a r t i c u l a r w e o b t a i n t h e c o n d i t i o n(3.9 ) Co = E [ C~ ] .T h i s m e a n s t h a t t h e o p t i m a l s t r a t e g y i s m e a n - s e l f - f i n a n c i n g : o n c e w e h a v e d e t e r -m i n e d t h e i n i t i a l v a l u e V o = C o , t h e a d d i t i o n a l c o s t C~ - C o i s a r a n d o m v a r i a b l ew i t h e x p e c t a ti o n E [ C t - C o ] = 0 . B y t h i s o p t i m a l t r a d i n g s t r a t e g y , t h e r e m a i n i n g


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HEDGING B Y SEQUENTIAL REGRESSION 35r is k is re d u c e d t o t h e m i n i m a l m e a n s q u a r e p r e d i c t i o n e r r o r(3 .10 ) Rmin = V a r [ H ] - ~ 2 . V a r [ X l ]

= V ar [ H ] (1 - ( p (H , X~ ) ) z )w h e r e p d e n o t e s t h e c o r r e l a t i o n c o e f f i c i e n t. T h i s v a l u e R m i . m a y b e v i e w e d a s t h ei n t r i n s i c r i s k o f t h e o p t i o n H . I t is t h i s in t r i n s i c r i s k , a n d n o t t h e a p r i o r i r i s km e a s u r e d b y th e v ar i a n ce V a r [ H ] o f H , o n w h i c h a n y a d j u s t m e n t o f t h e fa i rp r i c e V o b y a s u i ta b l e r is k p r e m i u m s h o u l d b e b a s e d .

I n a m o d e l w i t h a n i n t e r e s t r a t e r o n t h e S w i s s f r a n c a c c o u n t , w e w o u l d h a v et o r e p l a c e ( 3 . 5 ) b y

CI - Co = r/ l - (1 + r ) - T/o

= ( l + r ) . i + r V 0 - ~ - " l + r X o .T h e o p t i m a l c h o i c e o f ~ w o u l d s t il l b e ( 3 . 7 ); V o , h o w e v e r , i s t h e n g i v e n b y(3 .11 ) V o = - ~ . E l + r

T y p i c a l l y , t h e c r u c i a l q u a n t i t i e s ~ , V o a n d R m i , a l l d e p e n d o n t h e u n d e r l y i n gp r o b a b i l i t y m e a s u r e P . T h e i n t r i n s i c ri s k Rm in , a l t h o u g h s t r i c t l y l es s t h a n t h e ap r i o r i r i s k V a r [ H ] , i s s t il l s t r i c tl y p o s i t iv e a n d c a n n o t b e n e g l e c t e d . O n l y i n t h ef o l l o w i n g s c e n a ri o c a n w e e l i m i n a t e th e r i s k c o m p l e t e l y , i n a n a l o g y t o w h a t isp r o m i s e d b y th e B l a c k - S c h o l e s f o r m u l a .S u p p o s e t h a t o n l y t w o c a s e s a p p e a r w i t h p o s i ti v e p r o b a b i l i t y : E i t h e r th ee x c h a n g e r a t e g o e s u p t o s o m e l e v e l x + a n d t h e c o n t i n g e n t c l a i m H a s s u m e s a c o r -r e s p o n d i n g v a l u e h + , o r t h e e x c h a n g e r a t e g o es d o w n t o s o m e l ev e l x - a n d Ha s su m e s t h e v al u e h - . T h u s ,(3 .12) , a [ X t = x + , H = h ] = p

l a [ X l = x - , H = h - ] = 1 - pw i t h x - < (1 + r ) . Xo < x + a n d s o m e p E ( 0 , 1 ). T h i s a l l o w s u s t o d e t e r m i n e t w oc o n s t a n t s ~ a n d Vo s u c h t h a t(3 .13 ) P [ H = (1 + r ) . Vo + t~" (X~ - (1 + r ) - Xo )] = 1 .I n f a c t , (3 . 1 2 ) r e d u c e s ( 3 .1 3 ) t o t w o l i n e a r e q u a t i o n s f o r t h e t w o u n k n o w n s / ~ a n dV o , a n d t h e s o l u t i o n i s g i v e n b y

h + - h -(3 .14 ) ~ = x + - x -- - ( x + - ( l + r ) " x )V O = l + . h + . ( l + x + _ x: c - x - ~- h - . x + _ x _ "

T h u s , t h e r e is a l in e a r d e p e n d e n c e b e t w e e n t h e r a n d o m v a r i a b le s H a n dX ~ - (1 + r ) - X o , a n d t h e l i n e a r r e g r e ss i o n b e c o m e s p e r f e c t: Rmin = 0 . I n t h i s


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3 6 FOLLMER AND SCHWEIZERc a s e , t h e r e i s n o n e e d t o a d j u s t t h e f a i r p r i c e Vo b y a r is k p r e m i u m b e c a u s e t h er i s k h a s c o m p l e t e l y d i s a p p e a r e d . I n p a r t i c u l a r , w e h a v e(3.15) C , - Co = 0,i . e . t h e s t r a t e g y b e c o m e s self-financing. T h e r a n d o m v a r i a b l e T/1 i s n o w k n o w ni n a d v a n c e a n d c o i n c id e s w i t h t h e c o n s t a n t (1 + r ) " r/ o. F o r t h e s p e ci fi c v a l u e si n o u r i n t r o d u c t o r y e x a m p l e , w e r e c o v e r t h e s t r a t e g y w h i c h w a s d e s c r i b e d i nS e c t i o n 2 .

N o t e t h a t t h e o p t i m a l v a l u es ~ a n d Vo i n ( 3 . 1 4 ) d o n o t i n v o l v e t h e p r o b a b i l i t yp a r a m e t e r p ; t h e y a r e th e s a m e f o r a n y m e a s u r e / 5 w h i c h p r es e r v es t h e b i n a r ys t r u c t u r e ( 3 .1 2 ) w i t h s o m e p a r a m e t e r , 0 E ( 0 , I ) . I n p a r t i c u l a r , w e a r e f r e e t o s w i t c hf r o m P t o t h e m e a s u r e P * w i th

p * =s o t h a t(3 .16)

( l + r ) . x o - x -X + - - X -

E * [ x ~ ]x 0 = i_l + r j .T h u s , t h e d i sc o u n t e d e x c h a n g e r a t e b e h a v e s li ke a f a i r g a m e u n d e r P * . I n th i sn e w m o d e l , t h e f a i r p r i c e Vo c a n n o w b e c o m p u t e d d i r e c tl y , w i t h o u t g o i n gt h r o u g h t h e e x er c is e o f c o m p u t i n g ~ a n d Vo f r o m t w o l i n e a r e q u a t i o n s . I n f a c t ,(3 .15) impl ies

E * [ Vz - ~ " ( X t - ( I + r ) . X o ) ] = (1 + r ) . V o,a n d b y ( 3 . 3 ) a n d ( 3 . 1 6 ) w e o b t a i n(3 .17) Vo-- La + r J 'a s p r e s c r i b e d b y C h r . H u y g e n s a n d J . B e r n o u l l i .

T h e p r e c e d i n g d i s c u s s i o n o f t h e b i n a r y s c e n a r i o , v i e w e d a s a s p ec i a l c a s e o f t h eg e n e r a l tw o - p e r i o d m o d e l , is j u s t a n e l e m e n t a r y r e m a r k o n t h e l i n e ar r e g r e s s io np r o b l e m . B u t ( 3 .1 7 ) is t h e e x a ct a n a l o g u e t o t h e B l a c k - S c h o l e s f o r m u l a . I n b o t hc a s e s , t h e m o d e l i s comple t e i n t h e s e n s e t h a t a n y c o n t i n g e n t c l a i m H c a n b eg e n e r a t e d b y a s u i t a b l e s t r a t e g y a s in ( 3 . 13 ) . T h i s a l l o w s u s t o r e d u c e t h e r i s k t o0 , a n d t o c o m p u t e t h e v a l ue Vo b y a n a p p r o p r i a t e c h a n g e o f m e a s u r e .

4. HEDGING BY SEQUENTIAL REGRESSIONL e t u s n o w c o n s i d e r a m u l t i p e r i o d m o d e l w h e r e t h e e v o l u t i o n o f t h e e x c h a n g e r a t ei s g i v e n b y a s t o c h a s t i c p r o c e s s X k ( k = 0 , . . . , T ) o n s o m e p r o b a b i l i t y s p a c e( f l, ,~ ', P ) . Le t .~-k den o te the a - f i e ld o f even ts w hich a re obs e rva b le up to an d in -c l u d in g t im e k . W e a s s u m e t h a t X k is ~ k - m e a s u r a b l e a n d s q u a r e - i n t e g r a b l e . I no r d e r t o a v o i d c o m p l i c a t e d n o t a t i o n s , w e w o r k a g a i n w i t h o u t i n t e re s t r a t e s . T h is


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HEDGING BY SEQUENTIAL REGRESSION 37i s n o r e s t r i c t i o n s i n c e w e c a n a l w a y s s t a r t b y d i s c o u n t i n g t h e o r i g i n a l p r i c ep r o c e s s e s .

A n o p t i o n is d e s c ri b e d b y a s q u a r e - i n t e g r a b l e r a n d o m v a r i a b l e H E L2(p); f o re x a m p l e , a ca ll o p t i o n w i t h s tr i k e K w o u l d c o r r e s p o n d t o H = ( X T - - K ) + . At r a d i n g s t r a t e g y i s g i v e n b y t w o s t o c h a s t i c p r o c e s s e s ~ k ( k = 1, . . . , T ) a n d T/k( k = 0 , . .. , T ) . ~k i s t h e a m o u n t o f U S d o l l a r s h e l d i n p e r i o d k a n d h a s t o b e f i x eda t t h e b e g i n n i n g o f t h a t p e r i o d , i . e . w e a s s u m e t h a t( 4 .1 ) ~k i s . ~ ' k - t - m e a s u r a b l e ( k = 1 , . . . , T ) .T h e a m o u n t r/k o f S w i ss f r a n c s i n p e r i o d k c a n b e c h o s e n a t t h e e n d o f t h i s p e r i o d ,i . e . w e a s s u m e t h a t(4 .2 ) 7/k i s . ~k -m easu rab l e (k = 0 , . . . , T ) .F o r s u c h a t r a d i n g s t r a t e g y , t h e v a l u e o f t h e p o r t f o l i o a t t i m e k i s g i v e n b y(4 .3 ) V k = ~k " X k + */k (k = 1, . . . , T ) ,a n d Vo = r l0 . W e a d m i t o n l y s t r a te g i e s s u c h t h a t e a c h V k i s s q u a r e - i n t e g r a b l e a n ds u c h t h a t t h e c o n t i n g e n t c l a i m H i s p r o d u c e d i n th e e n d , i . e . w e r e q u i r e(4 .4 ) VT = H .D u e t o t h e f l ex i b i li ty a l l o w e d b y ( 4 . 2 ) , t h i s c a n a l w a y s b e a c h i e v e d b y a s u i t a b l ec h o i c e o f r / r . T h e c u m u l a t i v e c o s t a t t i m e k i s g i v e n b y

k(4 .5 ) Ck = V k - ~ a ~ i" A X jj = lw i t h A X i : = X j - X j - ~, a n d C o = Vo = 7 0. G o i n g b a c k w a r d s f r o m t i m e T , w e c a nn o w a p p l y t h e a r g u m e n t o f th e p r e c e d i n g s e c t io n s te p b y s t ep t o d e t e r m i n e o u rt r a d i n g s t r a t e g y r e c u r s i v e ly . S u p p o s e t h a t t h e r a n d o m v a r i a b l e s ~ t,+ 2 , . . ., / ~ r a n dr lk +~ , . . . , ~ r ( o r , e q u i v a l e n t l y , V k + ~ , . . . , V r ) h a v e a l r e a d y b e e n p r e s c r ib e d . A tt i m e k , w e w a n t t o c h o o s e f i rs t ~k . ~ a n d t h e n V k ( r e s p e c ti v e l y ~ k = V k - ~ k " X k )s u c h t h a t t h e c o n d i t i o n a l r i s k(4 .6 ) R k ~ E [ ( C k . t - C k ) 21 ,--~'k

= E [ ( V k + t - V k - ~ k + l " A X k + l ) 2 l , - ~ k ]i s m i n i m i z e d . N o t e t h a t t h e e x p e c t a t i o n i n ( 3 . 6 ) h a s n o w b e e n r e p l a c e d b y a c o n -d i t i o n a l e x p e c t a t i o n . I n a n a l o g y t o ( 3 . 9 ) , th i s im p l i e s( 4 . 7 ) C k = E [ C j , ; , t l ~ r k ] ,i . e . t h e c o s t p r o c e s s C k ( k = O , . . ., T ) is a m a r t i n g a l e . M o r e o v e r , t h e c o n d i t io n a lv e r s i o n s o f ( 3 . 7) a n d ( 3 .8 ) y i el d t h e r e c u r s i o n f o r m u l a e

. C o v . $ , _ ~ - j = k + 1(4.8 ) /ik = Var.~ ,_, [ A X k ]


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38 FOLLMER AND SCHWEIZERa n d(4 .9 )

l

I n p a r t i c u l a r , t h i s r e c u r s i o n l e a d s t o t h e f a i r p r i c e V0 = ~/0 o f o u r o p t i o n .T h e s t r u c t u re o f t h e o p t i m a l s t r a te g y b ec o m e s m u c h m o r e t r a n s p a r e n t i f P is

a m a r t i n g a l e m e a s u r e , i . e . i f X k ( k = O , . . ., T ) i s a m a r t i n g a l e u n d e r P :( 4 . 1 0 ) X ~ = E [ X k + t 1 .~ ' , ] ( k = 0 , . . . , T - I ) .B y ( 4 .7 ) a n d ( 4 . 5 ), i t f o l l o w s t h a t t h e v a l u e p r o c e s s V k ( k = 0 , . .. , T ) i s a l s o a m a r -t i n g a l e , h e n c e o f t h e f o r m(4 .1 I ) V k = E [ H ] ,~ -,1d u e t o ( 4 .4 ) . W e c a n n o w g i v e a m o r e d i r e c t c o n s t r u c t i o n o f th e o p t i m a l s t r a te g y .T o b e g i n w i t h , ( 4. I 1) d e t e r m i n e s ~ k = I I , - / j k " X k a s s o o n a s w e k n o w ~ k. I no r d e r t o c o m p u t e ~ k , w e u s e t h e f a c t t h a t H c a n b e w r i t t e n a s

T( 4 .1 2 ) H = V o + ~ ,, ~in " A X j + L ~j = lw h e r e L k ( k = O, . . ., T ) is a m a r t i n g a l e w h i c h i s o r t h o g o n a l t o X k ( k = O , . . ., T )i n t h e s e n s e t h a t( 4 .1 3 ) E [ A L ~ . A X , [ . ~ ' k - ~ ] = 0 .T h i s a l l o w s u s t o c o n c l u d e t h a t( 4 . 1 4 ) /jk = ~ ( k = 1 , . . . , T )i s t h e o p t i m a l h e d g i n g s t r a t e g y . I n f a c t , ( 4 . 1 1 ) a n d ( 4 . 1 2 ) i m p l y

"w

k( 4 .1 5 ) V k = V o + ~ ~ h , . A X j + L f f ,j = !h e n c e( 4 .1 6 ) R k - I = E [ ( A L ~ ) Z I , . ~ ' k - I ] + (~ k -- ~ kH )2 " E [ ( A X k ) 2 } J k - l ]d u e t o ( 4 . 1 3 ), a n d t h i s is m i n i m i z e d b y t h e c h o i c e o f ( 4 .1 4 ) .

N o w c o n s i d e r t h e v e r y s p e c ia l ca s e w h e r e H c a n a c t u a l l y b e g e n e r a t e d b y X k( k = 0 , . . . , T ) i n t h e s e n s e t h a t

T( 4 .1 7 ) H = V o + "~, b n " A X j ,j= li . e . w h e r e L r = 0 i n ( 4 . 1 2 ). T h e n ( 4 . 1 6 ) i m p l i e s R , = 0 ( k = 0 , . . . , T ) , h e n c e

C i , + 1 = C , = C o


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HEDGING BY SEQUENTIAL REGRESSION 3 9d u e t o ( 4 . 6 ) . T h u s , t h e o p t i m a l s t r a t e g y i s s e l f - f i n a n c i n g . B y ( 4 . 3 ) a n d ( 4 . 1 5 ) ,

,7,, = v , - ~ k" X ,= V ~ _ ~ + ~ k " ( X k - X ~ - t ) - ~ k " X k= V ~ _ ~ - ~ k" X k - l .

I n t h i s s p e c i a l c a s e , t h e p r o c e s s r/k ( k = 0 , . . . , T ) i s p r e d i c t a b l e , i . e .( 4 . 1 8 ) rlk i s . ~ r k - l - m e a s u r a b l e ( k = 1 , . . . , T ) .T h i s m e a n s t h a t a l l t h e i n g r e d i e n ts o f t h e t r a d i n g s t r a t e g y c a n a l r e a d y b e f ix e da t t h e b e g i n n i n g o f e a c h p e r i o d . N o t e a l s o t h a t t h e s tr a t e g y r e m a i n s t h e s a m e i fw e c h a n g e t h e m e a s u r e P t o a n y m e a s u r e / 5 w h i c h is e q u i v a l e n t t o P b e c a u s e t h isp r e s e r v e s t h e s t r u c t u r e ( 4 . 1 7 ). T h u s , t h e p r e c e d i n g d i s c u s s i o n i s v a l i d f o r a n ym o d e l P w h i c h c a n b e o b t a i n e d b y a n e q u i v a le n t c h a n g e o f m e a s u r e f r o m am a r t i n g a l e m o d e l P * s a t i s f y i n g ( 4 . 1 7 ) . I n s u c h a c a s e , t h e e x p l ic i t u se o f P *p e r m i t s a d i r e c t c o m p u t a t i o n o f t h e f a i r o p t i o n p r i c e . I n f a c t , ( 4 .1 7 ) im p l i e s

V0 = E * [ H ] ,i n a n a l o g y t o ( 3 . 1 7 ) a n d ( 2 . 5 ) .

5. OPTION TRADING AND STOCHASTIC CALCULUSL e t u s b r ie f ly c o m m e n t o n t h e e x te n s i o n o f o u r p r e v i o u s d i s c u s s io n t o ac o n t i n u o u s - t i m e s e t t in g w h e r e X t ( 0 ~ t ~< T ) is a s q u a r e - i n t e g r a b l e s e m i m a r -t i n g a l e o n s o m e p r o b a b i l i t y s p a c e ( fl , .~ r, p ) w i t h a r i g h t - c o n t i n u o u s f i l tr a t i o n 9 " t( 0 ~< t ~ T ) . W e c a n u se t h e s a m e c o n c e p t u a l a p p r o a c h t o t h e p r i c in g a n d h e d g i n go f o p t i o n s a s i n t h e d i sc r e t e -t i m e c a s e , b u t t h e t e c h n ic a l it i es a r e m u c h m o r ei n v o l v e d .H e r e a g a i n , t h e s i t u a t io n is m o s t t r a n s p a r e n t i n t h e c o m p l e t e c a s e w h e r e , i na n a l o g y t o (4 . 1 7 ), o u r s q u a r e - i n t e g r a b l e c o n t i n g e n t c l a im H c a n b e r e p r e s e n t e d a s

TH = V o + ( ~ H d X .( 5 . 1 ) J 0P - a l m o s t s u r e l y , i. e . a s a st o c h a s t i c i n t e g r a l o f s o m e p r e d i c t a b l e p r o c e s s ~ t( 0 ~< t ~< T ) w i t h r e s p e c t t o t h e b a s i c s e m i m a r t i n g a l e X t (0


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4 0 F O L L M E R A N D S C H W E I Z E R

sat isfiesC t = C o = V o (O


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HEDGING BY SEQUENTIAL REGRESSION 41i n t e n s i t y X ( t ) . T h i s m o d e l i s c o m p l e t e i n t h e s e n s e t h a t a n y s q u a r e - i n t e g r a b l e Hc a n b e r e p r e s e n t e d i n t h e f o r m

T( 6 . 2) H = V o + ~ ~ d X u .J oI n p a r ti c u la r , t h i s is tr u e f o r t h e r a n d o m v a r i a b l e H = ( S T - c ) a s s o c i a t e d t o as t o p - l o s s c o n t r a c t . N o w w e c o n s i d e r a n i n s u r a n c e s tr a t e g y g i v e n b y ~ ( 0 ~ t ~ T )a n d r/t ( 0 ~ t ~ T ) . H e r e , r /t i s t h e c a s h r e s e r v e h e l d a t t i m e t , a n d ~ t r e p r e s e n t sa p r o p o r t i o n a l r e i n s u r a n c e c o n t r a c t i n d u c i n g t h e ca p i t a l f l owI ' S '.(dSu-~6(u) du)= ~ . dX. .0 0W e t a k e ~ t = ~ ff , a n d w e d e t e r m i n e r/t = V t - ~ t" X t b y s e t t i n g

S '6 .3 ) Vt = Vo + ~ dX , , .oA s i n th e p r e v i o u s s e c t i o n , w e c o n c l u d e t h a t th i s st r a t eg y i s s e l f - f in a n c i n g a n d p r o -d u c e s t h e r a n d o m p a y m e n t H a t t h e t e r m i n a l t i m e T . T h e r e q u i r e d in i t ia l c o s t V oc a n b e c o m p u t e d d i r e c t l y a s t h e e x p e c t e d v a l u e

Vo = E * [ H ] - - E * [ ( S t - c ) ]i f w e s w i t c h f r o m P t o t h e n e w m o d e l P * w h e r e t h e p r o c e s s X , ( 0 ~< t ~< T ) is am a r t i n g a l e ; t h i s i s a c h i e v e d b y c h o o s i n g t h e r a te k * ( t ) = / ~ , (t ) . N o t e t h a t , i n c o m -p l e t e a n a l o g y t o o u r d i s c u s s i o n o f c u r r e n c y o p t i o n s , t h e fa i r p r e m i u m V o d o e s n o td e p e n d o n t h e a p r io r i ra t e X ( t ). F o r a m o r e d e t a i le d a c c o u n t s e e S O N D E R M A N N( 1 9 8 8 ) .

REFERENCESBACHELIER, L. "Th~orie de la Speculation", A n n . S c i . E c . No r m . S u p . H I - 1 7 (1900), 21-86.BERNOULLi, J. "Ars coniectandi" (1713).BLACK, F. and SCHOLES, M. "The Pricing o f Options and Corporate Liabilities", Jo u r n a l o f P o l i t ica lE c o n o m y 81 (1973), 637-659.Cox, J. C. and Ross, S. A. "The Valuation of Options for Alternative Stochastic Processes",Jo u r n a l o f F i n a n c i a l E co n o m i cs 3 (1976), 145-166.FOLLMER, H. and SONOERMANN, D. "Hedging of Non-Redundant Contingent Claims", in: HILOEN-BRAND, W. and MAS-COLELL, A. (eds.), Contributions to Mathemat ical Economics North-Holland (1986), 205-223.HARRISON, J. M. and PLISKA, S. R. "Martingales and Stochastic Integrals in t h e T h e o r y of Con-tinuous Trading", Stochas t ic Processes and their Appl ica t ions 11 (1981), 215-260.HARmSON, J. M. and PLISKA, S. R. "A Stochastic Calculus Model of Continuous Trading: C o m p l e t eMarkets", Stochas t ic Processes and their Appl ica t ions IS (! 983), 313-316.HUYGENS, CHR. "De ratiociniis in ludo aleae" (1657).ITO, K. "Multiple Wiener Integral", Jo u r n a l o f t h e M a t h em a t i ca l S o c i e t y o f Ja p a n 3 (1951),157-169.MERTON, R. C. "Theory of Rational Option Pric ing", B e ll J o u rn a l o f E c o n o m i c s a n d M a n a g e m e n tScience 4 (1973), 141-183SAMUELSON, P. A. "Ra tional Theory of Warrant Pricing", in: COOTNER, P. H. (ed.), "Th e Random

C h a r a c t e r of Stock Market Prices", MIT Press, Cambridge, Massachusetts (1964), 506-525.SCHWE[ZER, M. "Varianten der Black-Scholes-Formel", Diplomarbeit ETHZ, Ztirich (1984).


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42 F O L L M E R A N D S C H W E I Z E RSCHWEIZER, M. "He dgin g o f Opt ion s in a Genera l Semim ar t inga le M ode l" , D i s s . ET H Z n o. 8615,Ziirich (1988).SONDERMANN,D. "Re insurance in Arbi t rage-Free Marke t s " , d i s cus s ion paper no . B-82 , Univers i tyof Bonn (1988) .WIENER,N. "D i f fe rent i a l -Space" , Jou rna l o f M athem at ics and Phys ics 2 (1923), 131-174 ; repr in ted

in: WIENER, N . "Col l ec ted W ork s" , Volum e ! , MIT Pres s (1976).H A N S F O L L M E R , M A R T I N S C H W E I Z E Rl n s t i tu t f i i r An gew an d te Ma them at ik , Un ive rs itgi t Bon n , Wege le rs tr . 6 , D -53 00B o n n

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